Multiple extensions of a finite Euler's pentagonal number theorem and the Lucas formulas

Abstract: Motivated by the resemblance of a multivariate series identity and a finite analogue of Euler''s pentagonal number theorem, we study multiple extensions of the latter formula. In a different direction we derive a common extension of this multivariate series identity and two formulas of Lucas. Finally we give a combinatorial proof of Lucas’ formulas. [Copyright &y& Elsevier]